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Math Help - Volume using disk integration

  1. #1
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    Volume using disk integration

    Hello,
    I have to find the volume of the area between x=1 and y=(e^x)+1 rotated around both the x and y axis. The x axis worked fine, and I got about 10.83pi. However when I try to rotate the area around the y axis, I get the area between x=1 and x=ln(y-1) for 0<y<1, and when I attempt to integrate I get a large integral with ln(y-1) in it, and after subbing in 0 I get a negative number and therefore an error. What am I doing wrong? All help is appreciated
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  2. #2
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    Quote Originally Posted by Dudealadude View Post
    Hello,
    I have to find the volume of the area between x=1 and y=(e^x)+1 rotated around both the x and y axis. The x axis worked fine, and I got about 10.83pi. However when I try to rotate the area around the y axis, I get the area between x=1 and x=ln(y-1) for 0<y<1, and when I attempt to integrate I get a large integral with ln(y-1) in it, and after subbing in 0 I get a negative number and therefore an error. What am I doing wrong?
    disks and washers about the y-axis ...

    \displaystyle V = \pi \int_0^2 1^2 \, dy + \pi \int_2^{e+1} 1^2 - [\ln(y-1)]^2 \, dy<br />

    to check, method of cylindrical shells about the y-axis ...

    \displaystyle V = 2\pi \int_0^1 x(e^x + 1) \, dx
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  3. #3
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    one more way ...

    big cylinder - "scooped" out volume

    \displaystyle V = \pi \int_0^{e+1} 1^2 \, dy - \pi \int_2^{e+1} [\ln(y-1)]^2 \, dy
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  4. #4
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    Quote Originally Posted by Dudealadude View Post
    Hello,
    I have to find the volume of the area between x=1 and y=(e^x)+1 rotated around both the x and y axis. The x axis worked fine, and I got about 10.83pi. However when I try to rotate the area around the y axis, I get the area between x=1 and x=ln(y-1) for 0<y<1, and when I attempt to integrate I get a large integral with ln(y-1) in it, and after subbing in 0 I get a negative number and therefore an error. What am I doing wrong? All help is appreciated
    The region you describe (I put in bold above) is not bounded on the left. What limits of integration did you use to find the volume resulting from rotation about the x-axis?

    I assume you used 0≤x≤1.

    If this is the case, the the region being rotated is bounded by the coordinate axes as well as by x=1 and y=(e^x)+1.
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